To determine the stability of a fixed point for the given equation:

$\dot{a} = -i\nu a + a - |a|^2 a - i e$

we need to linearize it around a fixed point $a_0$. Here's the step-by-step procedure to do this and derive the characteristic equation:

1. Find the Fixed Points

First, we need to find the fixed points $a_0$ by setting $\dot{a} = 0$. The equation becomes:

$0 = -i\nu a_0 + a_0 - |a_0|^2 a_0 - i e$

This is a nonlinear equation in $a_0$, and solving it gives the fixed points.

2. Perturb Around the Fixed Point

Let's perturb the solution around the fixed point $a_0$:

$a(t) = a_0 + \delta a(t)$

where $\delta a(t)$ is a small perturbation. Substituting this into the original equation and linearizing (keeping only first-order terms in $\delta a$) will give the linearized equation.

3. Linearize the Equation

Substitute $a(t) = a_0 + \delta a(t)$ into the original equation:

$\dot{a} = -i\nu (a_0 + \delta a) + (a_0 + \delta a) - |a_0 + \delta a|^2 (a_0 + \delta a) - i e$

Expanding and linearizing around $a_0$, we get:

$\dot{\delta a} = \left.\frac{\partial \dot{a}}{\partial a}\right|_{a = a_0} \delta a$

Given that:

$\dot{\delta a} = (-i\nu + 1 - 2|a_0|^2)\delta a - a_0^2 \delta a^*$

Where $\delta a^*$ is the complex conjugate of $\delta a$.

4. Characteristic Equation

For stability, the eigenvalues $\lambda$ of the linearized equation determine the behavior of the perturbations. The equation governing the eigenvalues (the characteristic equation) can be derived from the linearized system.

Assume $\delta a(t) \propto e^{\lambda t}$, which leads to:

$\lambda \delta a = (-i\nu + 1 - 2|a_0|^2)\delta a - a_0^2 \delta a^*$

Taking the real and imaginary parts, this can be expressed as a system of linear equations, whose determinant gives the characteristic equation.

Given in the problem, the characteristic equation is provided as:

$\lambda^2 + (4R^2 - 2)\lambda + (1 - 3R^2)(1 - R^2) + \nu^2 = 0$

where $R = |a_0|$ is the amplitude of the fixed point.

5. Conclusion

The characteristic equation you've provided:

$\lambda^2 + (4R^2 - 2)\lambda + (1 - 3R^2)(1 - R^2) + \nu^2 = 0$

is obtained by solving the eigenvalue problem associated with the linearized system around the fixed point $a_0$. The coefficients of $\lambda$ and the constant term come from the linearization of the equation around the fixed point.

Summary

To determine the stability of the fixed point $a_0$, you linearize the equation around this point and solve for the eigenvalues using the characteristic equation. The sign of the real part of the eigenvalues will determine the stability of the fixed point.

Would you like more details on any of these steps or further clarification?

Related Questions

1. What are the steps to solve a nonlinear differential equation to find the fixed points?
2. How does the sign of the real part of eigenvalues determine the stability of a fixed point?
3. What is the significance of linearizing a nonlinear equation?
4. How can the characteristic equation be solved for a general nonlinear system?
5. What role does the perturbation method play in analyzing the stability of dynamical systems?

Tip

When analyzing the stability of a fixed point, always pay attention to the eigenvalues' real parts—positive real parts indicate instability, while negative real parts indicate stability.